Question

if alapa and beta are the zeros of the polynomial 2x²-5x+7 then find the value of alapa power -1 + beta power -1​

Answer 1

Given that

• Polynomial p(x)= 2x²-5x+7
• $\sf \alpha \: and \: \beta \: are \: the \: zeroes \: of \: polynomial \: p(x)$

To Find

• ${ \alpha }^{ - 1} + { \beta }^{ - 1}$

Solution

$\\P(x)= 2x²-5x+7$

$a=2 \ \ \ \ \ b= -5\ \ \ \ \ \ \ and ~~~~~~c=7$

$\\\gray{Sum \: of \: Zeros \: ( \alpha + \beta ) = \cfrac{ - b}{a}}$

$\gray {Sum \: of \: Zeros \: ( \alpha + \beta ) = \cfrac{ - ( - 5)}{2}}$

$\gray{ Sum \: of \: Zeros \: ( \alpha + \beta ) = \cfrac{ 5}{2}}$

$\\\gray {Product \: of \: zeros \: (\alpha \beta ) = \frac{c}{a}}$

$\gray{ Product \: of \: zeros \: (\alpha \beta ) = \cfrac{7}{2}}$

$\\\\→ \sf{ \alpha }^{ - 1} + { \beta }^{ - 1}= \cfrac{1}{ \alpha } + \cfrac{1}{ \beta }$

$→ \sf { \alpha }^{ - 1} + { \beta }^{ - 1}= \cfrac{ \beta + \alpha }{ \alpha \beta }$

$→ \sf { \alpha }^{ - 1} + { \beta }^{ - 1}= \dfrac{ \frac{5}{2} }{ \frac{7}{2} }$

$→\sf { \alpha }^{ - 1} + { \beta }^{ - 1}= \cfrac{5}{2} \times \cfrac{2}{7}$

$→ \bf { \alpha }^{ - 1} + { \beta }^{ - 1}= \cfrac{5}{7}$